Studying changes in the dynamical patterns in two physical models involving new Caputo operator

Abstract

Introduction: Studying phase changes in physical models enables researchers to adequately describe the dynamics arising from such systems. Objectives: This work investigates the effect of a new Caputo fractional operator on the dynamics in two physical models. Methods: The corresponding Riemann-Liouville fractional integral is presented. The main properties of the new Riemann-Liouville fractional integral are mentioned. In the case of the new Caputo fractional operator, new lemmas for derivatives of constant function, power function and its effect on the new fractional integral fractional integral are proved. In addition, an interpolation property is presented and proved. The new operator has a higher degree of freedom (new fractional parameter), which is used to produce changes in the dynamical patterns in some dynamical models. For example, the fractional modified autonomous Van der Pol-Duffing system exhibits variety of complex attractors such as one-band, double-band chaos and approximate periodic cycles as varying the new fractional parameter. In addition, the fractional Liu system shows variety of complex dynamics such as existences of double scroll, self-excited and hidden chaotic attractors. Conclusion: Both of the systems involving the new fractional operators exhibit variety of chaotic dynamics. It is also found in both systems that the new operator’s parameter arises chaotic attractors, while non-chaotic states are found in the systems’ counterparts with the classic fractional operator.

Keywords

• New Caputo fractional operator
• Generalization of Gamma function
• Van der Pol-Duffing model
• Liu model
• Chaos